Lectures on Duflo isomorphisms in Lie algebras and complex geometry [draft]

104 Pages·2011·0.755 MB·English

by  Damien Calaque, Carlo A. Rossi

Lectures on Duflo isomorphisms in Lie algebras
and complex geometry
Damien Calaque and Carlo A. Rossi
Contents
1 Lie algebra cohomology and the Duflo isomorphism 6
1.1 The original Duflo isomorphism . . . . . . . . . . . . . . . . . . . . 7
1.2 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Chevalley-Eilenberg cohomology . . . . . . . . . . . . . . . . . . . 11
1.4 The cohomological Duflo isomorphism . . . . . . . . . . . . . . . . 14
2 Hochschild cohomology and spectral sequences 15
2.1 Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Application: Chevalley-Eilenberg vs Hochschild cohomology . . . . 20
3 Dolbeault cohomology and the Kontsevich isomorphism 24
3.1 Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Atiyah and Todd classes . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Hochschild cohomology of a complex manifold . . . . . . . . . . . . 27
3.4 The Kontsevich isomorphism . . . . . . . . . . . . . . . . . . . . . 30
4 Superspaces and Hochschild cohomology 31
4.1 Supermathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Hochschild cohomology strikes back . . . . . . . . . . . . . . . . . 34
5 The Duflo-Kontsevich isomorphism for Q-spaces 38
5.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Application: proof of the Duflo Theorem . . . . . . . . . . . . . . . 40
5.3 Strategy of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Configuration spaces and integral weights 46
6.1 The configuration spaces C+ . . . . . . . . . . . . . . . . . . . . 46
n,m
6.2 Compactification of C and C+ a` la Fulton–MacPherson . . . . . 47
n n,m
6.3 Directed graphs and integrals over configuration spaces. . . . . . . 52
7 The map U and its properties 56
Q
7.1 The quasi-isomorphism property . . . . . . . . . . . . . . . . . . . 56
7.2 The cup product on polyvector fields . . . . . . . . . . . . . . . . . 60
7.3 The cup product on polydifferential operators . . . . . . . . . . . . 62
Contents 3
8 The map H and the homotopy argument 65
Q
8.1 The complete homotopy argument . . . . . . . . . . . . . . . . . . 65
8.2 Contribution to W2 of boundary components in Y . . . . . . . . . 67
Γ
8.3 Twisting by a supercommutative DG algebra . . . . . . . . . . . . 72
9 The explicit form of U 75
Q
9.1 Graphs contributing to U . . . . . . . . . . . . . . . . . . . . . . . 75
Q
9.2 U as a contraction. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Q
9.3 The weight of an even wheel . . . . . . . . . . . . . . . . . . . . . . 80
10 Fedosov resolutions 82
10.1 Bundles of formal fiberwise geometric objects . . . . . . . . . . . . 82
10.2 Resolutions of algebras . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.3 Fedosov differential . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.4 Fedosov resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
10.5 Proof of Theorem 3.6. . . . . . . . . . . . . . . . . . . . . . . . . . 89
A Deformation-theoretical interpretation of... 91
A.1 Cˇech cohomology: a (very) brief introduction . . . . . . . . . . . . 91
A.2 ThelinkbetweenCˇechandDolbeaultcohomology: DolbeaultThe-
orem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A.3 Twisted presheaves of algebras . . . . . . . . . . . . . . . . . . . . 95
Introduction
SincethefundamentalresultsbyHarish-Chandraandothers,itisnowwell-known
thatthealgebraofinvariantpolynomialsonthedualofaLiealgebraofaparticular
type (solvable [18], simple [24] or nilpotent) is isomorphic to the center of the cor-
respondinguniversalenvelopingalgebra. Thisfactwasgeneralizedtoanarbitrary
finite-dimensional real Lie algebra by M. Duflo in 1977 [19]. His proof is based on
Kirillov’s orbits method that parametrizes infinitesimal characters of unitary irre-
duciblerepresentationsofthecorrespondingLiegroupintermsofco-adjointorbits
(see e.g. [28]). This isomorphism is called the Duflo isomorphism. It happens to
beacompositionofthewell-knownPoincar´e-Birkhoff-Wittisomorphism(whichis
only an isomorphism at the level of vector spaces) with an automorphism of the
space of polynomials (which descends to invariant polynomials), whose definition
involves the power series j(x):=sinh(x/2)/(x/2).
In 1997 Kontsevich [29] proposed another proof, as a consequence of his con-
struction of deformation quantization for general Poisson manifolds. Kontsevich’s
approach has the advantage to work also for Lie super-algebras and to extend the
Duflo isomorphism to a graded algebra isomorphism on the whole cohomology.
Theinversepowerseriesj(x)−1 =(x/2)/sinh(x/2)alsoappearsinKontsevich’s
claim that the Hochschild cohomology of a complex manifold is isomorphic as an
algebra to the cohomology ring of holomorphic polyvector fields on this manifold.
Wecansummarizetheanalogybetweenthetwosituationsintothefollowingtable:
Lie algebra Complex geometry
symmetric algebra sheaf of algebra of holomorphic
polyvector fields
universal enveloping algebra sheaf of algebra of holomorphic
polydifferential operators
taking invariants taking global holomorphic sections
Chevalley-Eilenberg cohomology sheaf cohomology
These lecture notes provide a self-contained proof of the Duflo isomorphism
and its complex geometric analogue in a unified framework, and gives in particu-
laraunifyingexplanationofthereasonwhytheseriesj(x)anditsinverseappear.
TheproofisstronglybasedonKontsevich’soriginalidea,butactuallydiffersfrom
it (the two approaches are related by a conjectural Koszul type duality recently
pointed out in [39], this duality being itself a manifestation of Cattaneo-Felder
constructions for the quantization of a Poisson manifold with two coisotropic sub-
Contents 5
manifolds [12]).
Note that the series j(x) also appears in the wheeling theorem by Bar-Natan,
Le and Thurston [4] which shows that two spaces of graph homology are isomor-
phicasalgebras(seealso[31]foracompletelycombinatorialproofofthewheeling
theorem, based on Alekseev and Meinrenken’s proof [1, 2] of the Duflo isomor-
phism for quadratic Lie algebras). Furthermore this power series also shows up in
various index theorems (e.g. Riemann-Roch theorems).
Throughout these notes we assume that k is a field with char(k) = 0. Unless
otherwise specified, algebras, modules, etc... are over k.
Each chapter consists (more or less) of a single lecture.
Acknowledgements
Theauthorsthanktheparticipantsofthelecturesfortheirinterestandexcitement.
They are responsible for the very existence of these notes, as well as for improve-
mentoftheirquality. ThefirstauthorisgratefultoG.Felderwhoofferedhimthe
opportunity to give this series of lectures. He also thanks M. Van den Bergh for
hiskindcollaborationin[9]andmanyenlightingdiscussionsaboutthisfascinating
subject. His research is fully supported by the European Union thanks to a Marie
Curie Intra-European Fellowship (contract number MEIF-CT-2007-042212).
1 Lie algebra cohomology and the Duflo
isomorphism
Let g be a finite dimensional Lie algebra over k. In this chapter we state the
Duflo theorem and its cohomological extension. We take this opportunity to in-
troducestandardnotionsofhomologicalalgebraanddefinethecohomologytheory
associated to Lie algebras, which is called Chevalley-Eilenberg cohomology.
Preliminaries: tensor, symmetric, and universal envelopping
algebras For k-vector space V we define the tensor algebra T(V) of V as the
vector space
T(V):= V⊗n (V⊗0 =k by convention)
n≥0
M
equipped with the product given by the concatenation. It is a graded algebra,
whose subspace of homogeneous elements of degree n is Tn(V):=V⊗n.
The symmetric algebra of V, which we denote by S(V), is the quotient of the
tensor algebra T(V) by its two-sided ideal generated by
v⊗w−w⊗v (v,w ∈V).
Since the previous relations are homogeneous, then S(V) inherits a grading from
the one on T(V).
Finally, if V =g, one can define the universal enveloping algebra U(g) of g as
the quotient of the tensor algebra T(g) by its two-sided ideal generated by
x⊗y−y⊗x−[x,y] (x,y ∈V),
where [x,y] denotes the Lie bracket between x and y. As the relations are not
homogeneous, the universal enveloping algebra only inherits a filtration from the
grading on the tensor algebra.
Notation. Dealing with non-negatively graded vector spaces, we will use the
symbol to denote the corresponding degree completions. Namely, if M is a
graded k-vector space, then
b M := Mn
n≥0
Y
c
is the set of formal series
m(n), (m(n) ∈Mn).
n≥0
X
1.1 The original Duflo isomorphism 7
1.1 The original Duflo isomorphism
The Poincar´e-Birkhoff-Witt theorem
Recall the Poincar´e-Birkhoff-Witt (PBW) theorem: the symmetrization map
I : S(g) −→ U(g)
PBW
1
x ···x 7−→ x ···x ,
1 n n! σ1 σn
σX∈Sn
is an isomorphism of filtered vector spaces, which further induces an isomorphism
of the corresponding graded algebras S(g)→Gr U(g) .
(cid:0) (cid:1)
Let us write ∗ for the associative product on S(g) defined as the pullback of
the multiplication on U(g) through I . For any two homogeneous elements
PBW
u,v ∈S(g), u∗v =uv+l.o.t. (where l.o.t. stands for “lower order terms”).
I is obviously NOT an algebra isomorphism, unless g is abelian (since
PBW
S(g) is commutative while U(g) is not).
Remark 1.1. There are different proofs of the PBW Theorem: standard proofs
may be found in [16], to which we refer for more details. More conceptual proofs,
involving Koszul duality between quadratic algebras, may be found in [6, 37]. A
proof of the PBW Theorem stemming from Kontsevich’s Deformation Quantiza-
tion may be found in [39, 8].
Geometric meaning of the PBW theorem
We consider a connected, simply connected Lie group G with corresponding
Lie algebra g.
Then S(g) can be viewed as the algebra of distributions on g supported at the
origin 0 with (commutative) product given by the convolution with respect to the
(abelian) additive group law on g.
In the same way U(g) can be viewed as the algebra of distributions on G
supported at the origin e with product given by the convolution with respect to
the group law on G.
One sees that I is nothing but the transport of distributions through the
PBW
exponential map exp:g→G (recall that it is a local diffeomorphism). The expo-
nential map is obviously Ad-equivariant. In the next paragraph we will translate
this equivariance in algebraic terms.
g-module structure on S(g) and U(g)
On the one hand there is a g-action on S(g) obtained from the adjoint action
ad of g on itself, extended to S(g) by Leibniz’ rule: for any x,y ∈g and n∈N∗,
ad (yn)=n[x,y]yn−1.
x
8 1 Lie algebra cohomology and the Duflo isomorphism
On the other hand there is also an adjoint action of g on U(g): for any x ∈ g
and u∈U(g),
ad (u)=xu−ux.
x
It is an easy exercise to verify that ad ◦I =I ◦ad for any x∈g.
x PBW PBW x
Therefore I restricts to an isomorphism (of vector spaces) from S(g)g to
PBW
the center Z(Ug)=U(g)g of Ug.
Now we have commutative algebras on both sides. Nevertheless, I is not
PBW
yet an algebra isomorphism. Theorem 1.3 below is concerned with the failure of
this map to preserve the product.
Duflo element J
WedefineanelementJ ∈S(g∗)(thesetofformalpowerseriesong)asfollows:
b 1−e−adx
J(x):=det .
ad
x
(cid:16) (cid:17)
It can be expressed as a formal power series w.r.t. c :=tr((ad)k).
k
Let us explain what this means. Recall that ad is the linear map g→End(g)
defined by ad (y) = [x,y] (x,y ∈ g). Therefore ad ∈ g∗ ⊗ End(g) and thus
x
(ad)k ∈Tk(g∗)⊗End(g). Consequently tr((ad)k)∈Tk(g∗)and weregardit as an
element of Sk(g∗) through the projection T(g∗)→S(g∗).
Notation. Hereandbelow,foravectorspaceV wedenotebyEnd(V)thealgebra
of endomorphisms of V, and by V∗ the vector space of linear forms on V.
Claim 1.2. c is g-invariant.
k
Here the g-module structure on S(g∗) is the coadjoint action on g∗ extended
by Leibniz’ rule.
Proof. Let x,y ∈g. Then
n n
hy·c ,xni = −hc , xi[y,x]xn−i−1i=− tr(adiad adn−i−1)
k k x [y,x] x
i=1 i=1
X X
n
= − tr(adi[ad ,ad ]adn−i−1)=−tr([ad ,adn])=0
x y x x y x
i=1
X
This proves the claim. 2
The Duflo isomorphism
Observethatanelementξ ∈g∗ actsonS(g)asaderivationasfollows: forany
x∈g
ξ·xn =nξ(x)xn−1.
1.2 Cohomology 9
By extension an element (ξ)k ∈Sk(g∗) acts as follows:
(ξ)k·xn =n···(n−k+1)ξ(x)kxn−k.
This way the algebra S(g∗) acts on S(g).1 Moreover, one sees without difficulty
that S(g∗)g acts on S(g)g. We have:
b
Theobrem 1.3 (Duflo,[19]). IPBW ◦ J1/2· defines an isomorphism of algebras
S(g)g →U(g)g.
The proof we will give in these lectures is based on deformation theory and
homologicalalgebra,followingthedeepinsightofM.Kontsevich[29](seealso[38]).
Remark 1.4. c is a derivation of S(g), thus exp(c ) defines an algebra auto-
1 1
morphism of S(g). Therefore one can obviously replace J by the modified Duflo
element
eadx/2−e−adx/2
J(x)=det .
ad
(cid:18) x (cid:19)
e
Remark 1.5. It has been proved by Duflo that, for any finite-dimensional Lie
algebra g, the trace of odd powers of the adjoint representation of g acts trivially
on S(g)g. In [30, Theorem 8], Kontsevich states that such odd powers act as
derivations on S(g)g, where now g may be a finite-dimensional graded Lie algebra
(see Chapter 4). It is not known if, for a finite-dimensional graded Lie algebra
g, the traces of odd powers of the adjoint representation act trivially on S(g)g: if
not,theywouldprovideanon-trivialincarnationoftheactionoftheGrothendieck–
Teichmueller group on deformation quantization.
1.2 Cohomology
Our aim is to show that Theorem 1.3 is the degree zero part of a more general
statement. For this we need a few definitions.
Definition 1.6. 1. A DG vector space is a Z-graded vector space C• =⊕n∈ZCn
equipped with a graded linear endomorphism d : C → C of degree one (i.e.
d(Cn)⊂Cn+1) such that d◦d=0. d is called the differential.
2. A DG (associative) algebra is a DG vector space (A•,d) equipped with an
associative product which is graded (i.e. Ak · Al ⊂ Ak+l) and such that d is
1Thisactioncanberegardedastheactionofthealgebraofdifferentialoperatorswithconstant
coefficientsong∗ (ofpossiblyinfiniteorder)ontofunctionsong∗.
10 1 Lie algebra cohomology and the Duflo isomorphism
a graded derivation of degree 1: for homogeneous elements a,b ∈ A d(a·b) =
d(a)·b+(−1)|a|a·d(b).
3. Let (A•,d) be a DG algebra. A DG A-module is a DG vector space (M•,d)
equipped with an A-module structure which is graded (i.e. Ak·Ml ⊂Mm+l) and
suchthatdsatisfiesd(a·m)=d(a)·m+(−1)|a|a·d(m)forhomogeneouselements
a∈A, m∈M.
4. A morphism of DG vector spaces (resp. DG algebras, DG A-modules) is
a degree preserving linear map that intertwines the differentials (resp. and the
products, the module structures).
DG vector spaces are also called cochain complexes (or simply complexes) and
differentials are also known as coboundary operators. Recall that the cohomology
of a cochain complex (C•,d) is the graded vector space H•(C,d) defined by the
quotient ker(d)/im(d):
{c∈Cn|d(c)=0} {n-cocycles}
Hn(C,d):= = .
{b=d(a)|a∈Cn−1} {n-coboundaries}
Anymorphismofcochaincomplexesinducesadegreepreservinglinearmapatthe
level of cohomology. The cohomology of a DG algebra is a graded algebra.
Example 1.7 (Differential-geometric induced DG algebraic structures). Let M
be a differentiable manifold. Then the graded algebra of differential forms Ω•(M)
equipped with the de Rham differential d=d is a DG algebra. Recall that for
dR
any ω ∈Ωn(M) and v ,...,v ∈X(M)
0 n
n
d(ω)(u ,··· ,u ) := (−1)iu ω(u ,...,u ,...,u )
0 n i 0 i n
i=0
X (cid:0) (cid:1)
+ (−1)i+jω([u ,u ],u ,...,u ,...,ub,...,u ).
i j 0 i j n
0≤i<j≤n
X
b b
In local coordinates (x1,...,xn), the de Rham differential reads d=dxi ∂ . The
∂xi
corresponding cohomology is denoted by H• (M).
dR
For any C∞ map f : M → N one has a morphism of DG algebras given by the
pullback of forms f∗ :Ω•(N)→Ω•(M).
Let E →M be a vector bundle and recall that a connection ∇ on M with values
in E is given by the data of a linear map ∇:Γ(M,E)→Ω1(M,E) such that for
any f ∈C∞(M) and s∈Γ(M,E) one has ∇(fs)=d(f)s+f∇(s). Observe that
it extends in a unique way to a degree one linear map ∇:Ω•(M,E)→Ω•(M,E)
such that for any ξ ∈ Ω•(M) and s ∈ Ω•(M,E), ∇(ξs) = d(ξ)s+(−1)|ξ|ξ∇(s).
Thereforeiftheconnectionisflat(whichisbasicallyequivalenttotherequirement
that ∇◦∇ = 0) then Ω•(M,E) becomes a DG Ω(M)-module. Conversely, any
differential∇thatturnsΩ(M,E)inaDGΩ(M)-moduledefinesaflatconnection.